Generally, among the methods for investigating dynamic characteristics of mechanical systems, the system response to periodic input is investigated. In vibration systems, this is called forced vibration; whereas, in an automatic control system, it is called the frequency response.
This method is widely used for understanding the dynamic characteristics of the vehicle, and the vehicle response to a periodic steering input is examined here. This is important, as an on-board driver can feel the vehicle’s lateral acceleration and yaw rate responses to steering input very well.
First, the lateral acceleration of the vehicle’s center of gravity to a periodic steering input,d, can be written as follows by multiplying bys2 on both sides ofEqn (3.87)0 and substituting sẳj2pf:
G€ydðj2pfị ẳGyd€ð0ị 1 ð2pfị2Ty2ỵj2pfTy1
1 ð2pfị2=u2nỵj2pf2z=un (3.92) wherefis the frequency of the periodic steering, andjẳ ffiffiffiffiffiffiffi
p1 .
From this equation, the lateral acceleration gainG€yd and the phase angle:Gyd€ toward steering angle are as follows:
G€ydẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2yþQ2y R2yþS2y vu
ut Gyd€ð0ị (3.93)
:G€ydẳtan1 Qy=Py
tan1 Sy=Ry
(3.94)
where
Pyẳ1 ð2pfị2Ty2 Qyẳ2pf Ty1 Ryẳ1 ð2pfị2=u2n Syẳ2pf2z=un
Next, the vehicle yaw rate,r, response,Grdðj2pfị, to periodic steering,d, can be written by substitutingsẳj2pfintoEqn (3.78)0:
Grdðj2pfị ẳGrdð0ị 1ỵj2pfTr 1 ð2pfị2=u2nỵj2pf2z=un
(3.95) From this equation, the yaw rate gain,Grd, and the phase angle,:Grd, to steering input are as follows:
Grdẳ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P2rþQ2r R2rþS2r s
Grdð0ị (3.96)
:Grdẳtan1ðQr=Prịtan1ðSr=Rrị (3.97)
where
Prẳ1; Qrẳ2pfTr
Rrẳ1 ð2pfị2=u2n Srẳ2pf2z=un
In particular, the investigation of the yaw rate response to a periodic steering is very common in the study of the vehicle’s inherent dynamic characteristics. The yaw rate response as expressed byEqn (3.95)has a general form as shown inFigure 3.30. When the steering frequency is small, the yaw rate to steering gain is almost constant. As the steering frequency becomes larger, the US vehicle gain reaches a peak at a certain frequency and then decreases. The OS and NS vehicles do not have a peak, and their gain decreases with steering frequency. Furthermore, the phase lag is around zero at low steering frequencies, but it increases with frequency for all three steer char- acteristics. This tendency is more obvious for the vehicle with OS characteristics. For the US vehicle, the peak in the gain happens when the vehicle transient response to a fixed steering is oscillating. This peak becomes greater as the damping ratio,z, reduces. When the vehicle exhibits US characteristics, the peak will become larger with increasing traveling speed,V. The frequency where the peak occurs is nearly the same as the vehicle natural frequency,un.
Figure 3.31is a calculated example of the responses of yaw rate and lateral acceleration to a periodic steering for a small passenger car. From this figure, it is clear that with higher traveling speed, the motion phase lag, especially in the lateral acceleration response, becomes larger at higher frequencies. Furthermore, because the vehicle is US, a gain peak occurs at high traveling speed in the yaw rate response, and the vehicle transient response is oscillatory with insufficient damping.
Here, the relation between the frequency responses of lateral acceleration and yaw rate will be considered. The transfer function of the lateral acceleration to the steering input is derived in this section from equations of motion with fixed coordinates on the ground,Eqn (3.21)0and (3.22)0. There is another way to derive the lateral acceleration transfer function using equations of motion with the fixed coordinates on the vehicle,Eqns (3.75) and (3.76). Based on these, the lateral
f
gainphase
FIGURE 3.30
Conceptual diagram of yaw rate frequency response.
acceleration is expressed byVðb_ỵrị. A side-slip rate intervenes between the lateral acceleration response and the yaw rate.
Therefore, usingEqn (3.77)0and (3.78)0, it is possible to describe the lateral acceleration transfer function as follows:
G€ydðsị ẳVsbðsị dðsị ỵVrðsị
dðsị
ẳVGbdð0ịs 1þTbs 1þ2zsu
n þus22 n
ỵVGrdð0ị 1ỵTrs 1þ2zsu
nþus22 n
ẳ 1 1þAV2
V2 l
lr
Vs2lmlKfV
rsþ2lKI
rs2 1þ2zsu
nþus22 n
þ 1 1þAV2
V2 l
1þm2llKfV
rs 1þ2zsu
n þus22 n
ẳ 1 1þAV2
V2 l
1þVlrsþ2lKI
rs2 1þ2zsu
n þus22 n
ẳG€ydð0ị1ỵTy1sỵTy2s2 1þ2zsu
n þus22 n
The coefficient ofsin the numerator of the yaw rate transfer function,mlfV/(2lKr), is elimi- nated by the same term, which is a negative part of steady-state response of side-slip angle to steering input in the numerator of the side-slip transfer function. Only the termlr/Vremains in the numerator of the lateral acceleration transfer function as a coefficient ofs, and this rapidly de- creases with the vehicle speed.
The coefficient ofsin the numerator of the transfer function, in general, has a lead effect and compensates for the response delay caused by the coefficient ofsands2in the denominator. The lateral acceleration response to steering has a smaller value of the coefficient ofsin the numerator
Freqency (Hz)
Gain (1/s)
Freqency (Hz) Gain (m/s2/deg)
Lateral acceleration Yaw rate
60 (km/h)
100 (km/h) 140 (km/h)
60 (km/h) 100 (km/h)
140 (km/h)
60 (km/h)
100 (km/h) 60 (km/h)
100 (km/h) 140 (km/h)
140 (km/h)
Freqency (Hz)
)ged(elgnaesahP
Freqency (Hz)
)ged(elgnaesahP
FIGURE 3.31
Yaw rate and lateral acceleration frequency response.
compared with that of the yaw rate, especially at high vehicle speed. This is partly why there is a significantly larger delay in the phase lag of the lateral acceleration compared with that of the yaw rate at high speed, as shown inFigure 3.31. The larger delay in lateral acceleration is due to the side-slip response acting in opposite to the steering angle at higher vehicle speeds. This is a very important part of the basic nature of the vehicle dynamics and is attributed to the intervention of vehicle side-slip motion between lateral acceleration and yaw rate.
Example 3.7
Calculate the yaw rate frequency responses of the vehicles with US, NS, and OS characteristics, respec- tively, at the vehicle speedVẳ120 km/h using Matlab-Simulink and confirm the effects of the steer characteristics on the yaw rate frequency response shown previously inFigure 3.30.
Solution
The parameters of the US vehicle are the same as in Example 3.6, and the cornering stiffness for the NS vehicle is set asKfẳ68.15 kN/rad andKrẳ46.85 kN/rad; for the OS vehicle, it isKfẳ72.5 kN/rad and Krẳ42.5 kN/rad. The parameters of the vehicle and the sweep-type sine wave are set as in Figure E3.7(a). The simulation program of the vehicle response to the sweep-type sine wave steering input is shown inFigure E3.7(b), andFigure E3.7(c)is a result of the simulation. After finishing the vehicle response simulation to the sweep-type of sine wave steering input, the simulated data is saved as shown inFigure E3.7(d), and the yaw rate frequency response to steering input is calculated applying the Fourier Transformation to the time histories simulated, as shown in Figure E3.7(e). A result is inFigure E3.7(f), andFigure E3.7(g)shows the summarized calculation results comparing the effects of the vehicle steer characteristics on the yaw rate frequency response.
FIGURE E3.7(a)
FIGURE E3.7(b)
FIGURE E3.7(c)
FIGURE E3.7(e)
Freqency (Hz)
Gain (1/s)
Freqency (Hz) Gain (m/s 2/deg)
Lateral acceleration Yaw rate
US NS
OS
V=120(km/h)
US NS
OS
)ged(elgnaesahP
OS NS
US
Freqency (Hz) Freqency (Hz)
Phase angle (deg)
OS NS
US
V=120(km/h)
FIGURE E3.7(g)